1. Field of the Invention
The present invention concerns a method and a device to determine a relative position (angular position) by means of a resolver, in particular at high rotation speeds.
2. Description of the Prior Art
Resolvers are used in engineering, among other things, to detect the relative position of actuated or non-actuated pivot points, motors and the like. In robotics, the evaluation of the resolver signals is accorded particular importance since they decisively influence the performance and the positioning precision of the robot.
A known design of a resolver includes two stator windings offset by 90° that enclose a rotatably supported rotor with a rotor winding. Other resolver designs employ two windings arranged offset from one another and activated by a stator winding, or make use of variable magnetic resistance (as known from EP 0 877 464 A2).
The rotor of the resolver is activated by a reference sine signal of the form U(t)=UR·sin(2π·f·t) with the amplitude UR and the frequency f that induces voltages of different amplitude in the stator windings S1 and S2 depending on the rotor position. If ρ designates the defined angle of the rotor as shown in FIG. 1, the voltageUS1(t)=CTF·UR·sin(2π·f·t+φR)·cos ρ=US1,Amp·sin(2π·f·t+φR),   (1)results for S1 and the voltageUS2(t)=CTF·UR·sin(2π·f·t+φR)·sin ρ=US2,Amp·sin(2π·f·t+φR).   (2)results for S2.
The two induced voltages are thus theoretically identical in frequency f and phase but can be shifted in phase by φR relative to the reference sine with which the rotor winding is activated, wherein CTF designates the transmission factor. The angle ρ of the resolver can thus be determined by
                    ρ        =                              arc            ⁢                                                  ⁢                          tan              ⁡                              (                                                                                                    C                        TF                                            ·                                              U                        R                                            ·                      sin                                        ⁢                                                                                  ⁢                    ρ                                                                                                      C                        TF                                            ·                                              U                        R                                            ·                      cos                                        ⁢                                                                                  ⁢                    ρ                                                  )                                              =                      arc            ⁢                                                  ⁢                          tan              ⁡                              (                                                      U                                                                  S                        ⁢                                                                                                  ⁢                        2                                            ,                      Amp                                                                            U                                                                  S                        ⁢                                                                                                  ⁢                        1                                            ,                      Amp                                                                      )                                                                        (        3        )            
An optimally precise determination of the amplitudes US1,Amp and US2,Amp is thus important for a good position signal.
In addition to the conventional sampling of the voltages US1(t), US2(t) in the range of their extremes (in which US(t)≈US,Amp applies), difficulties exist due to the sensitivity regarding the sampling point in time. From U.S. Pat. No. 5,241,268 it is known to implement a Fourier transformation of the voltages US1(t), US2(t) in order to determine the amplitudes US1,Amp and US2,Amp (and thus the rotor angle ρ) more reliably and precisely.
For this purpose, the voltage signals US1(t), US2(t) are sampled equidistantly. The calculation of the amplitude of these N time-discrete sample values then ensues by means of (for example) discrete Fourier transformation, which transforms a time signal into a frequency range. The complex Fourier coefficients â=(â0, . . . , âN-1) are calculated from the time-discrete sample values a=(a0, . . . , aN-1) according to:
                                                        a              ^                        k                    =                                                    1                N                            ⁢                                                ∑                                      j                    =                    0                                                        N                    -                    1                                                  ⁢                                                                  ⁢                                                                            ⅇ                                                                        -                          ⅈ                                                ⁢                                                                                                  ⁢                        2                        ⁢                        π                        ⁢                                                  jk                          N                                                                                      ·                                          a                      j                                                        ⁢                                                                          ⁢                  for                  ⁢                                                                          ⁢                  k                                                      =            0                          ,                  .                                          .                                          .                ⁢                                  ,                  N          -          1                                    (        4        )            
The Fourier coefficient âk at the frequency of the resolver signal contains the amplitude 2·|âk| as well as the phase ∠(âk) of the sampled signal. Which of the coefficients corresponds to the exciter signal depends on the number of full waves across which the Fourier transformation is calculated. If it is one full wave, the first coefficient is calculated and Equation (4) is simplified as:
                                          a            ^                    1                =                              1            N                    ⁢                                    ∑                              j                =                0                                            N                -                1                                      ⁢                                                  ⁢                                          ⅇ                                                      -                    ⅈ                                    ⁢                                                                          ⁢                  2                  ⁢                  π                  ⁢                                      j                    N                                                              ·                              a                j                                                                        (        5        )            
Given two full waves, the second coefficient is accordingly calculated, given three full waves the third coefficient etc.
Equation (5) can also be represented with separate real part and imaginary part with the aid of the Euler formula eiθ=cos θ+i sin θ:
                                          a            ^                    1                =                                            1              N                        ⁢                                          ∑                                  j                  =                  0                                                  N                  -                  1                                            ⁢                                                cos                  ⁡                                      (                                          2                      ⁢                                              π                        ·                                                  j                          /                          N                                                                                      )                                                  ·                                  a                  j                                                              -                      ⅈ            ⁢                          1              N                        ⁢                                          ∑                                  j                  =                  0                                                  N                  -                  1                                            ⁢                                                sin                  ⁡                                      (                                          2                      ⁢                                              π                        ·                                                  j                          /                          N                                                                                      )                                                  ·                                  a                  j                                                                                        (        6        )            
The Fourier coefficient from Equation (6) is to be calculated for both US1(t) and US2(t).
To solve Equation (3) a complex division
                                          z                          S              ⁢                                                          ⁢              1                                            z                          S              ⁢                                                          ⁢              2                                      =                                                            a                ^                                            1                ,                                  S                  ⁢                                                                          ⁢                  1                                                                                    a                ^                                            1                ,                                  S                  ⁢                                                                          ⁢                  2                                                              =                                                                      x                                      S                    ⁢                                                                                  ⁢                    1                                                  +                                  ⅈy                                      S                    ⁢                                                                                  ⁢                    1                                                                                                x                                      S                    ⁢                                                                                  ⁢                    2                                                  +                                  ⅈy                                      S                    ⁢                                                                                  ⁢                    2                                                                        =                                                                                                      x                                              S                        ⁢                                                                                                  ⁢                        1                                                              ⁢                                          x                                              S                        ⁢                                                                                                  ⁢                        2                                                                              +                                                            y                                              S                        ⁢                                                                                                  ⁢                        1                                                              ⁢                                          y                                              S                        ⁢                                                                                                  ⁢                        2                                                                                                                                  x                                          S                      ⁢                                                                                          ⁢                      2                                        2                                    +                                      y                                          S                      ⁢                                                                                          ⁢                      2                                        2                                                              +                              ⅈ                ⁢                                                                                                    x                                                  S                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              y                                                  S                          ⁢                                                                                                          ⁢                          1                                                                                      -                                                                  x                                                  S                          ⁢                                                                                                          ⁢                          1                                                                    ⁢                                              y                                                  S                          ⁢                                                                                                          ⁢                          2                                                                                                                                                x                                              S                        ⁢                                                                                                  ⁢                        2                                            2                                        +                                          y                                              S                        ⁢                                                                                                  ⁢                        2                                            2                                                                                                                              (        7        )            of the two (complex) Fourier coefficients zS1, zS2 that result for the two voltage signals US1(t), US2(t) can be taken into account. The absolute value of this complex number, i.e. the real part according to Equation (7), corresponds to the quotient of the amplitudes to be used in Equation (3).
However, the determined position disadvantageously becomes increasingly more imprecise with increasing rotor rotation sped. It has been shown that the determined deviation is approximately linearly dependent on h rotation speed and approximately sinusoidal relative to the phase displacement of the resolver. This is shown in FIG. 6 in which the deviation is plotted in increments over the phase displacement of the resolver (increasing from left to right in FIG. 6) and the rotation speed of the resolver (increasing from front to back in FIG. 6). In addition to this position deviation, the jitter of the position also increases massively, as this is shown by way of example in FIG. 7 in which peak-to-peak values of the jitter are plotted over the phase displacement of the resolver (increasing from left to right in FIG. 7) and the rotation speed of the resolver (increasing from front to back in FIG. 7).